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%I #40 Oct 25 2023 09:29:37
%S 0,1,13,3,26,5,39,7,52,9,65,11,78,13,91,15,104,17,117,19,130,21,143,
%T 23,156,25,169,27,182,29,195,31,208,33,221,35,234,37,247,39,260,41,
%U 273,43,286,45,299,47,312,49,325,51,338,53,351,55,364,57,377,59,390,61,403,63,416,65,429,67,442,69
%N Multiples of 13 and odd numbers interleaved.
%C Partial sums give the generalized 17-gonal numbers (A303305).
%C More generally, the partial sums of the sequence formed by the multiples of m and the odd numbers interleaved, give the generalized k-gonal numbers, with m >= 1 and k = m + 4.
%C a(n) is also the length of the n-th line segment of the rectangular spiral whose vertices are the generalized 17-gonal numbers.
%H Colin Barker, <a href="/A317313/b317313.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,0,-1).
%F a(2n) = 13*n, a(2n+1) = 2*n + 1.
%F From _Colin Barker_, Jul 29 2018: (Start)
%F G.f.: x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2).
%F a(n) = 2*a(n-2) - a(n-4) for n>3. (End)
%F Multiplicative with a(2^e) = 13*2^(e-1), and a(p^e) = p^e for an odd prime p. - _Amiram Eldar_, Oct 14 2023
%F Dirichlet g.f.: zeta(s-1) * (1 + 11/2^s). - _Amiram Eldar_, Oct 25 2023
%t Table[{13n, 2n + 1}, {n, 0, 35}] // Flatten (* or *)
%t CoefficientList[Series[(x^3 + 13 x^2 + x)/(x^2 - 1)^2, {x, 0, 69}], x] (* or *)
%t LinearRecurrence[{0, 2, 0, -1}, {0, 1, 13, 3}, 70] (* _Robert G. Wilson v_, Jul 26 2018 *)
%o (PARI) a(n) = if(n%2==0, return((n/2)*13), return(n)) \\ _Felix Fröhlich_, Jul 26 2018
%o (PARI) concat(0, Vec(x*(1 + 13*x + x^2) / ((1 - x)^2*(1 + x)^2) + O(x^60))) \\ _Colin Barker_, Jul 29 2018
%Y Cf. A008595 and A005408 interleaved.
%Y Column 13 of A195151.
%Y Sequences whose partial sums give the generalized k-gonal numbers: A026741 (k=5), A001477 (k=6), zero together with A080512 (k=7), A022998 (k=8), A195140 (k=9), zero together with A165998 (k=10), A195159 (k=11), A195161 (k=12), A195312 k=13), A195817 (k=14), A317311 (k=15), A317312 (k=16).
%Y Cf. A303305.
%K nonn,easy,mult
%O 0,3
%A _Omar E. Pol_, Jul 25 2018
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